Sorting is one of the most fundamental algorithms in computer science. Recently, Learned Sorts, which use machine learning to improve sorting speed, have attracted attention. While existing studies show that Learned Sort is experimentally faster than classical sorting algorithms, they do not provide theoretical guarantees about its computational complexity. We propose PCF Learned Sort, a theoretically guaranteed Learned Sort algorithm. We prove that the expected complexity of PCF Learned Sort is $O(n \log \log n)$ under mild assumptions on the data distribution. We also confirm experimentally that PCF Learned Sort has a computational complexity of $O(n \log \log n)$ on both synthetic and real datasets. This is the first study to theoretically support the experimental success of Learned Sort, and provides evidence for why Learned Sort is fast.

翻译：排序是计算机科学中最基础的算法之一。近年来，利用机器学习提升排序速度的"学习型排序"算法引起了广泛关注。尽管现有研究表明，学习型排序算法在实验中比经典排序算法更快，但这些研究并未提供其计算复杂度的理论保证。我们提出了PCF Learned Sort——一种具有理论保证的学习型排序算法。我们证明，在数据分布的温和假设下，PCF Learned Sort的期望复杂度为$O(n \log \log n)$。此外，我们通过实验验证，PCF Learned Sort在合成数据集和真实数据集上均实现了$O(n \log \log n)$的计算复杂度。这是首个为学习型排序算法的实验成功提供理论支撑的研究，并解释了学习型排序为何能够实现高效运行。